Estimation

Author

Josie Athens

Published

February 15, 2024

2 The Normal distribution

Example

A normal curve was calculated from the diastolic blood pressures of 500 men (mean 82 mm Hg, standard deviation 10 mm Hg). What is the probability of finding a man with a DBP > 72 mm Hg if we assume a normal distribution?

Before we answer this problem, we need to think about our answer. By definition, the mean value is at the centre, so the area under the curve (AUC) AUC = 50% to each side. As the value of 72 is less than the mean (82), we already know that the solution has to be greater than 50%.

The cdf is the cumulative distribution function, i.e., the area under the curve to the left of our asking point. In this example, we want to know the AUC to the right of 72.

 1 - cdf(Normal(82, 10), 72) |> r3
0.841

Let’s report as percentage:

(1-cdf(Normal(82, 10), 72)) * 100 |> r3
84.134
Exercise

What is the probability of finding a man with a DBP < 65 mm Hg in our current example?

Code
cdf(Normal(82, 10), 65) * 100 |> r3
4.457

What would be the value of DBP which would contain 97.5% of the population (to the left of the curve)?

This is the opposite problem than the previous one. Now we know the probability and would like to know the value that would contain that probability. For finding quantiles in the normal curve, we use the quantile command.

quantile(Normal(82, 10), 0.975) |> r3
101.6
Exercise

What would be value of DBP which would contain 2.5% of the population (to the left of the curve)?

Code
quantile(Normal(82, 10), 0.025) |> r3
62.4

So, the limits that would contain 95% of the population around the mean are:

r3.(reference_range(82, 10))
2-element Vector{Float64}:
  62.4
 101.6

Which \(z\)-value would contain 97.5% of the standard normal curve?

quantile(Normal(0, 1), 0.975) |> r3
1.96

Let’s check on the distribution of height from the WCGS dataset.

wcgs = read_rds("data/wcgs.rds") |> rcopy
wcgs |> schema
┌──────────┬───────────────────────────────┬────────────────────────────────────
│ names    │ scitypes                      │ types                             ⋯
├──────────┼───────────────────────────────┼────────────────────────────────────
│ id       │ Count                         │ Int64                             ⋯
│ age      │ Count                         │ Int64                             ⋯
│ height   │ Continuous                    │ Float64                           ⋯
│ weight   │ Continuous                    │ Float64                           ⋯
│ sbp      │ Count                         │ Int64                             ⋯
│ dbp      │ Count                         │ Int64                             ⋯
│ chol     │ Union{Missing, Count}         │ Union{Missing, Int64}             ⋯
│ beh_pat  │ Multiclass{4}                 │ CategoricalValue{String, UInt32}  ⋯
│ ncigs    │ Count                         │ Int64                             ⋯
│ dib_pat  │ Multiclass{2}                 │ CategoricalValue{String, UInt32}  ⋯
│ chd      │ Multiclass{2}                 │ CategoricalValue{String, UInt32}  ⋯
│ type_chd │ Multiclass{4}                 │ CategoricalValue{String, UInt32}  ⋯
│ time     │ Count                         │ Int64                             ⋯
│ arcus    │ Union{Missing, Multiclass{2}} │ Union{Missing, CategoricalValue{S ⋯
│ smoker   │ Multiclass{2}                 │ CategoricalValue{String, UInt32}  ⋯
└──────────┴───────────────────────────────┴────────────────────────────────────
                                                                1 column omitted
estat(@select(wcgs, :height), ["Height (cm)"])
1×6 DataFrame
Row Variable n Median Mean SD CV
String Int64 Float64 Float64 Float64 Float64
1 Height (cm) 3154 177.8 177.235 6.423 0.036
qq_plot(wcgs.height, ylab =  "Height (cm)")
Theoretical quantiles 150 160 170 180 190 200 210 150.0 152.5 155.0 157.5 160.0 162.5 165.0 167.5 170.0 172.5 175.0 177.5 180.0 182.5 185.0 187.5 190.0 192.5 195.0 197.5 200.0 202.5 205.0 207.5 210.0 150.0 150.2 150.4 150.6 150.8 151.0 151.2 151.4 151.6 151.8 152.0 152.2 152.4 152.6 152.8 153.0 153.2 153.4 153.6 153.8 154.0 154.2 154.4 154.6 154.8 155.0 155.2 155.4 155.6 155.8 156.0 156.2 156.4 156.6 156.8 157.0 157.2 157.4 157.6 157.8 158.0 158.2 158.4 158.6 158.8 159.0 159.2 159.4 159.6 159.8 160.0 160.2 160.4 160.6 160.8 161.0 161.2 161.4 161.6 161.8 162.0 162.2 162.4 162.6 162.8 163.0 163.2 163.4 163.6 163.8 164.0 164.2 164.4 164.6 164.8 165.0 165.2 165.4 165.6 165.8 166.0 166.2 166.4 166.6 166.8 167.0 167.2 167.4 167.6 167.8 168.0 168.2 168.4 168.6 168.8 169.0 169.2 169.4 169.6 169.8 170.0 170.2 170.4 170.6 170.8 171.0 171.2 171.4 171.6 171.8 172.0 172.2 172.4 172.6 172.8 173.0 173.2 173.4 173.6 173.8 174.0 174.2 174.4 174.6 174.8 175.0 175.2 175.4 175.6 175.8 176.0 176.2 176.4 176.6 176.8 177.0 177.2 177.4 177.6 177.8 178.0 178.2 178.4 178.6 178.8 179.0 179.2 179.4 179.6 179.8 180.0 180.2 180.4 180.6 180.8 181.0 181.2 181.4 181.6 181.8 182.0 182.2 182.4 182.6 182.8 183.0 183.2 183.4 183.6 183.8 184.0 184.2 184.4 184.6 184.8 185.0 185.2 185.4 185.6 185.8 186.0 186.2 186.4 186.6 186.8 187.0 187.2 187.4 187.6 187.8 188.0 188.2 188.4 188.6 188.8 189.0 189.2 189.4 189.6 189.8 190.0 190.2 190.4 190.6 190.8 191.0 191.2 191.4 191.6 191.8 192.0 192.2 192.4 192.6 192.8 193.0 193.2 193.4 193.6 193.8 194.0 194.2 194.4 194.6 194.8 195.0 195.2 195.4 195.6 195.8 196.0 196.2 196.4 196.6 196.8 197.0 197.2 197.4 197.6 197.8 198.0 198.2 198.4 198.6 198.8 199.0 199.2 199.4 199.6 199.8 200.0 200.2 200.4 200.6 200.8 201.0 201.2 201.4 201.6 201.8 202.0 202.2 202.4 202.6 202.8 203.0 203.2 203.4 203.6 203.8 204.0 204.2 204.4 204.6 204.8 205.0 205.2 205.4 205.6 205.8 206.0 206.2 206.4 206.6 206.8 207.0 207.2 207.4 207.6 207.8 208.0 208.2 208.4 208.6 208.8 209.0 209.2 209.4 209.6 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162.05332340848537160.02 161.96853709309423160.02 161.8810029518457160.02 161.79051950504498160.02 161.69686179951205160.02 161.59977757798194160.02 161.498982629176160.02 161.39415509798908160.02 161.28492846250737160.02 161.17088278301077160.02 161.05153368417075160.02 160.92631832429072160.02 160.79457730132802160.02 160.65553099053741160.02 160.50824811303227160.02 160.3516032446675160.02 160.18421821906153158.73993341788207 160.0043794608319157.48 159.8099182591634157.48 159.59803196841287157.48 159.3650071173205157.48 159.10577145964436156.20396005072922 158.8131293231301154.94 158.4763649594716154.94 158.0784561485728153.66637603043756 157.58980217647817152.4 156.9513689975434152.4 156.01387749349013152.4 154.1106646735312152.4 h,j,k,l,arrows,drag to pan i,o,+,-,scroll,shift-drag to zoom r,dbl-click to reset c for coordinates ? for help ? 150 160 170 180 190 200 152 154 156 158 160 162 164 166 168 170 172 174 176 178 180 182 184 186 188 190 192 194 196 198 200 152.4 152.6 152.8 153.0 153.2 153.4 153.6 153.8 154.0 154.2 154.4 154.6 154.8 155.0 155.2 155.4 155.6 155.8 156.0 156.2 156.4 156.6 156.8 157.0 157.2 157.4 157.6 157.8 158.0 158.2 158.4 158.6 158.8 159.0 159.2 159.4 159.6 159.8 160.0 160.2 160.4 160.6 160.8 161.0 161.2 161.4 161.6 161.8 162.0 162.2 162.4 162.6 162.8 163.0 163.2 163.4 163.6 163.8 164.0 164.2 164.4 164.6 164.8 165.0 165.2 165.4 165.6 165.8 166.0 166.2 166.4 166.6 166.8 167.0 167.2 167.4 167.6 167.8 168.0 168.2 168.4 168.6 168.8 169.0 169.2 169.4 169.6 169.8 170.0 170.2 170.4 170.6 170.8 171.0 171.2 171.4 171.6 171.8 172.0 172.2 172.4 172.6 172.8 173.0 173.2 173.4 173.6 173.8 174.0 174.2 174.4 174.6 174.8 175.0 175.2 175.4 175.6 175.8 176.0 176.2 176.4 176.6 176.8 177.0 177.2 177.4 177.6 177.8 178.0 178.2 178.4 178.6 178.8 179.0 179.2 179.4 179.6 179.8 180.0 180.2 180.4 180.6 180.8 181.0 181.2 181.4 181.6 181.8 182.0 182.2 182.4 182.6 182.8 183.0 183.2 183.4 183.6 183.8 184.0 184.2 184.4 184.6 184.8 185.0 185.2 185.4 185.6 185.8 186.0 186.2 186.4 186.6 186.8 187.0 187.2 187.4 187.6 187.8 188.0 188.2 188.4 188.6 188.8 189.0 189.2 189.4 189.6 189.8 190.0 190.2 190.4 190.6 190.8 191.0 191.2 191.4 191.6 191.8 192.0 192.2 192.4 192.6 192.8 193.0 193.2 193.4 193.6 193.8 194.0 194.2 194.4 194.6 194.8 195.0 195.2 195.4 195.6 195.8 196.0 196.2 196.4 196.6 196.8 197.0 197.2 197.4 197.6 197.8 198.0 198.2 150 200 Height (cm)
Figure 1: QQ-plot of height in males from the WCGS dataset.

2.1 Log-normal distribution

Some variables, which distribution is right-skewed, can be approximate to a normal distribution after a log-transformation.

Let’s load the vector IgM from the R package ISwR. For consistency, we will use a data frame.

igm = rcopy(R"ISwR::IgM");
IgM = DataFrame(igm = igm);

Descriptive statistics:

summarize(IgM, :igm)
    | Obs | Mean  | Std. Dev. |  Min  |  Max  
----------------------------------------------
igm | 298 | 0.803 |     0.469 | 0.100 | 4.500 
estat(
    @select(IgM, {:igm}),
    ["IgM (mg/ml)"]
)
1×6 DataFrame
Row Variable n Median Mean SD CV
String Int64 Float64 Float64 Float64 Float64
1 IgM (mg/ml) 298 0.7 0.803 0.469 0.585

Let’s compare the skweness of height and IgM:

skewness(wcgs.height) |> r3, skewness(IgM.igm) |> r3
(-0.199, 2.554)

Clearly, our variable is positively (right) skewed. Let’s check on the corresponding QQ-plot.

qq_plot(IgM.igm, ylab = "IgM (mg/ml)")
Theoretical quantiles -1 0 1 2 3 4 5 -1.00 -0.75 -0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90 -0.88 -0.86 -0.84 -0.82 -0.80 -0.78 -0.76 -0.74 -0.72 -0.70 -0.68 -0.66 -0.64 -0.62 -0.60 -0.58 -0.56 -0.54 -0.52 -0.50 -0.48 -0.46 -0.44 -0.42 -0.40 -0.38 -0.36 -0.34 -0.32 -0.30 -0.28 -0.26 -0.24 -0.22 -0.20 -0.18 -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.84 1.86 1.88 1.90 1.92 1.94 1.96 1.98 2.00 2.02 2.04 2.06 2.08 2.10 2.12 2.14 2.16 2.18 2.20 2.22 2.24 2.26 2.28 2.30 2.32 2.34 2.36 2.38 2.40 2.42 2.44 2.46 2.48 2.50 2.52 2.54 2.56 2.58 2.60 2.62 2.64 2.66 2.68 2.70 2.72 2.74 2.76 2.78 2.80 2.82 2.84 2.86 2.88 2.90 2.92 2.94 2.96 2.98 3.00 3.02 3.04 3.06 3.08 3.10 3.12 3.14 3.16 3.18 3.20 3.22 3.24 3.26 3.28 3.30 3.32 3.34 3.36 3.38 3.40 3.42 3.44 3.46 3.48 3.50 3.52 3.54 3.56 3.58 3.60 3.62 3.64 3.66 3.68 3.70 3.72 3.74 3.76 3.78 3.80 3.82 3.84 3.86 3.88 3.90 3.92 3.94 3.96 3.98 4.00 4.02 4.04 4.06 4.08 4.10 4.12 4.14 4.16 4.18 4.20 4.22 4.24 4.26 4.28 4.30 4.32 4.34 4.36 4.38 4.40 4.42 4.44 4.46 4.48 4.50 4.52 4.54 4.56 4.58 4.60 4.62 4.64 4.66 4.68 4.70 4.72 4.74 4.76 4.78 4.80 4.82 4.84 4.86 4.88 4.90 4.92 4.94 4.96 4.98 5.00 -5 0 5 2.17780378897074963.6030201342282338 2.0092520330843442.6010067114093998 1.92395149192708412.352516778523494 1.86482161518765472.1511744966442965 1.81892034030587762.1 1.78108985380910672.0518456375838925 1.74872846925020482.0 1.72033262642020042.0 1.69495174832291221.905704697986573 1.67194509049922551.8 1.65085944164062531.8 1.6313619590754321.7538590604026867 1.61320071219931731.7000000000000002 1.59618023061211311.7000000000000002 1.58014567707608761.654865771812081 1.5649722260499221.6 1.55055771471516431.555536912751677 1.53681742357331961.5000000000000002 1.5236802846188771.5000000000000002 1.51108607151301141.5000000000000002 1.4989832807227281.5000000000000002 1.48732750868081.5000000000000002 1.47608019143596941.4575503355704713 1.46520761348595661.4000000000000001 1.45468011941260131.4000000000000001 1.44447148032291571.4000000000000001 1.43455837987780971.4000000000000001 1.4249199937168241.4000000000000001 1.41553764255891211.4000000000000001 1.40639450396290981.4000000000000001 1.39747537119287381.4000000000000001 1.38876645021069331.3605704697986596 1.38025518775779531.3000000000000003 1.37193012496191911.3000000000000003 1.36378077203584461.3000000000000003 1.35579750051012881.3000000000000003 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0.181120274739552940.30000000000000004 0.171481888578567050.30000000000000004 0.161568788133460980.30000000000000004 0.151360149043775150.30000000000000004 0.140832654970419860.30000000000000004 0.129960077020407130.30000000000000004 0.11871275977557660.30000000000000004 0.107056987733648690.30000000000000004 0.094954196943365150.30000000000000004 0.08235998383749910.30000000000000004 0.069222844883056970.30000000000000004 0.055482553741212140.30000000000000004 0.0410680424064547860.30000000000000004 0.02589459138028960.30000000000000004 0.009860037844263480.30000000000000004 -0.0071604437429405590.30000000000000004 -0.0253216906190553150.30000000000000004 -0.044819173184248660.30000000000000004 -0.065904822042849310.2468120805369127 -0.088911479866535690.2 -0.11429235796382420.2 -0.142688200793828050.2 -0.175049585352729520.2 -0.212880071849500860.2 -0.258781346731277730.2 -0.317911223470707660.14916107382550337 -0.403211764627967040.1 -0.57176352051437530.1 h,j,k,l,arrows,drag to pan i,o,+,-,scroll,shift-drag to zoom r,dbl-click to reset c for coordinates ? for help ? 1 2 3 0 5 0 4 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.20 0.21 0.22 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.40 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.50 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.60 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.70 0.71 0.72 0.73 0.74 0.75 0.76 0.77 0.78 0.79 0.80 0.81 0.82 0.83 0.84 0.85 0.86 0.87 0.88 0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18 1.19 1.20 1.21 1.22 1.23 1.24 1.25 1.26 1.27 1.28 1.29 1.30 1.31 1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.39 1.40 1.41 1.42 1.43 1.44 1.45 1.46 1.47 1.48 1.49 1.50 1.51 1.52 1.53 1.54 1.55 1.56 1.57 1.58 1.59 1.60 1.61 1.62 1.63 1.64 1.65 1.66 1.67 1.68 1.69 1.70 1.71 1.72 1.73 1.74 1.75 1.76 1.77 1.78 1.79 1.80 1.81 1.82 1.83 1.84 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 1.93 1.94 1.95 1.96 1.97 1.98 1.99 2.00 2.01 2.02 2.03 2.04 2.05 2.06 2.07 2.08 2.09 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 2.20 2.21 2.22 2.23 2.24 2.25 2.26 2.27 2.28 2.29 2.30 2.31 2.32 2.33 2.34 2.35 2.36 2.37 2.38 2.39 2.40 2.41 2.42 2.43 2.44 2.45 2.46 2.47 2.48 2.49 2.50 2.51 2.52 2.53 2.54 2.55 2.56 2.57 2.58 2.59 2.60 2.61 2.62 2.63 2.64 2.65 2.66 2.67 2.68 2.69 2.70 2.71 2.72 2.73 2.74 2.75 2.76 2.77 2.78 2.79 2.80 2.81 2.82 2.83 2.84 2.85 2.86 2.87 2.88 2.89 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 3.00 3.01 3.02 3.03 3.04 3.05 3.06 3.07 3.08 3.09 3.10 3.11 3.12 3.13 3.14 3.15 3.16 3.17 3.18 3.19 3.20 3.21 3.22 3.23 3.24 3.25 3.26 3.27 3.28 3.29 3.30 3.31 3.32 3.33 3.34 3.35 3.36 3.37 3.38 3.39 3.40 3.41 3.42 3.43 3.44 3.45 3.46 3.47 3.48 3.49 3.50 3.51 3.52 3.53 3.54 3.55 3.56 3.57 3.58 3.59 3.60 3.61 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 IgM (mg/ml)
Figure 2: QQ-plot of IgM concentration in the blood against the standard normal distribution.

And now, with a log-transformation:

qq_plot(log.(IgM.igm), ylab = "log (IgM)")
Theoretical quantiles -3 -2 -1 0 1 2 -3.0 -2.8 -2.6 -2.4 -2.2 -2.0 -1.8 -1.6 -1.4 -1.2 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 -3.00 -2.98 -2.96 -2.94 -2.92 -2.90 -2.88 -2.86 -2.84 -2.82 -2.80 -2.78 -2.76 -2.74 -2.72 -2.70 -2.68 -2.66 -2.64 -2.62 -2.60 -2.58 -2.56 -2.54 -2.52 -2.50 -2.48 -2.46 -2.44 -2.42 -2.40 -2.38 -2.36 -2.34 -2.32 -2.30 -2.28 -2.26 -2.24 -2.22 -2.20 -2.18 -2.16 -2.14 -2.12 -2.10 -2.08 -2.06 -2.04 -2.02 -2.00 -1.98 -1.96 -1.94 -1.92 -1.90 -1.88 -1.86 -1.84 -1.82 -1.80 -1.78 -1.76 -1.74 -1.72 -1.70 -1.68 -1.66 -1.64 -1.62 -1.60 -1.58 -1.56 -1.54 -1.52 -1.50 -1.48 -1.46 -1.44 -1.42 -1.40 -1.38 -1.36 -1.34 -1.32 -1.30 -1.28 -1.26 -1.24 -1.22 -1.20 -1.18 -1.16 -1.14 -1.12 -1.10 -1.08 -1.06 -1.04 -1.02 -1.00 -0.98 -0.96 -0.94 -0.92 -0.90 -0.88 -0.86 -0.84 -0.82 -0.80 -0.78 -0.76 -0.74 -0.72 -0.70 -0.68 -0.66 -0.64 -0.62 -0.60 -0.58 -0.56 -0.54 -0.52 -0.50 -0.48 -0.46 -0.44 -0.42 -0.40 -0.38 -0.36 -0.34 -0.32 -0.30 -0.28 -0.26 -0.24 -0.22 -0.20 -0.18 -0.16 -0.14 -0.12 -0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.32 0.34 0.36 0.38 0.40 0.42 0.44 0.46 0.48 0.50 0.52 0.54 0.56 0.58 0.60 0.62 0.64 0.66 0.68 0.70 0.72 0.74 0.76 0.78 0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02 1.04 1.06 1.08 1.10 1.12 1.14 1.16 1.18 1.20 1.22 1.24 1.26 1.28 1.30 1.32 1.34 1.36 1.38 1.40 1.42 1.44 1.46 1.48 1.50 1.52 1.54 1.56 1.58 1.60 1.62 1.64 1.66 1.68 1.70 1.72 1.74 1.76 1.78 1.80 1.82 1.84 1.86 1.88 1.90 1.92 1.94 1.96 1.98 2.00 -3 0 3 1.23825372051487981.2495216748660536 1.04191617218101130.9551586402331744 0.94255381885343330.8534464737325524 0.87367636948335670.7657437285693817 0.82020825977252780.7419373447293773 0.77614140961132730.7184427522518151 0.73844524399400520.6931471805599453 0.70536834049455880.6931471805599453 0.67580341929883580.6434721723387198 0.64900410908872330.5877866649021191 0.62444248657607240.5877866649021191 0.60173083990469430.561413235697446 0.58057570765709210.5306282510621705 0.56074939609647210.5306282510621705 0.54207154303386070.5032657959134776 0.52439674541116990.47000362924573563 0.50760599884781540.44130781028356 0.491600617818136240.40546510810816455 0.47629781874095010.40546510810816455 0.46162744695279490.40546510810816455 0.447529508552417350.40546510810816455 0.43395228003518460.40546510810816455 0.420850840174353150.3761778656816575 0.408185915459993940.336472236621213 0.39592296177201870.336472236621213 0.38403142637890740.336472236621213 0.372484149238162640.336472236621213 0.3612568730891180.336472236621213 0.35032783936731380.336472236621213 0.339677452448617470.336472236621213 0.329287998763464040.336472236621213 0.31914341032365510.3072518113592604 0.309229064463299230.2623642644674913 0.299531613312630560.2623642644674913 0.29003883784079730.2623642644674913 0.280739522323141370.2623642644674913 0.27162334588386170.2623642644674913 0.26268078839017970.2623642644674913 0.253903048469223960.23268399870263795 0.245281971813494310.1823215567939548 0.236809988257386080.1823215567939548 0.2284800563627940.1823215567939548 0.220285614459234560.1823215567939548 0.21222053725322610.1823215567939548 0.204279097260501420.15180917795027596 0.196455930429124130.09531017980432493 0.188746005416357220.09531017980432493 0.181144596060999090.09531017980432493 0.173647256658760240.09531017980432493 0.16624979970352150.09531017980432493 0.1589482758038010.09531017980432493 0.151738955523097460.09531017980432493 0.144618312926083850.09531017980432493 0.13758301064101820.09531017980432493 0.130629886272933260.09531017980432493 0.123755940022919060.09531017980432493 0.11695832338662760.09531017980432493 0.110234328820454970.09531017980432493 0.103581380277141870.09531017980432493 0.096997024523984570.09531017980432493 0.090478923166842470.06700497539934876 0.084024845311791160.0 0.077632660803857110.0 0.071300333988883470.0 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Figure 3: QQ-plot of log(IgM) against the standard normal distribution.

If we calculate the reference range, the result makes no sense, as, of course, we cannot have negative concentrations.

r3.(reference_range(mean(IgM.igm), std(IgM.igm)))
2-element Vector{Float64}:
 -0.117
  1.723

We should make the calculation in the log scale and then back transform the results.

igm_log = log.(IgM.igm)
r3.(exp.(reference_range(mean(igm_log), std(igm_log))))
2-element Vector{Float64}:
 0.238
 2.031

2.2 Random Sampling

When we take a sample, we calculate a statistic, an estimate of a parameter in the original population where the sample was initially taken.

This process, of statistical inference is reliable and robust if the process is not biased. One way to avoid bias is through random sampling. The function sample allows us to take random samples from variables.

To have the same results, we will set up a seed before the sampling.

using StableRNGs
rng = StableRNG(576);

Look at the statistics from random samples with decreasing sample size.

estat(
  @select(wcgs[sample(rng, 1:nrow(wcgs), 100), :], {:height})
)
1×6 DataFrame
Row Variable n Median Mean SD CV
Symbol Int64 Float64 Float64 Float64 Float64
1 height 100 177.8 176.251 6.537 0.037
estat(
  @select(wcgs[sample(rng, 1:nrow(wcgs), 50), :], {:height})
)
1×6 DataFrame
Row Variable n Median Mean SD CV
Symbol Int64 Float64 Float64 Float64 Float64
1 height 50 177.8 177.546 6.099 0.034
estat(
  @select(wcgs[sample(rng, 1:nrow(wcgs), 25), :], {:height})
)
1×6 DataFrame
Row Variable n Median Mean SD CV
Symbol Int64 Float64 Float64 Float64 Float64
1 height 25 177.8 177.19 5.74 0.032
estat(
  @select(wcgs[sample(rng, 1:nrow(wcgs), 10), :], {:height})
)
1×6 DataFrame
Row Variable n Median Mean SD CV
Symbol Int64 Float64 Float64 Float64 Float64
1 height 10 175.26 174.752 7.063 0.04

So, even with a sample as small size as \(n=10\), the sample statistics are a good estimate of the corresponding statistics in the population.

3 Confidence Intervals

Reference range and confidence intervals are not the same.

  • The reference range is about the limits that would contain 95% of the observations in a sample or population.
  • The confidence interval is about the limits that would contain a given proportion (most of the times, 95%; sometimes, 99%) of the true value of a given parameter (e.g., the mean) in a population.

For the calculation of the reference range, we used about two standard deviations (~ 1.96) around the mean (as we assumed a normal distribution). For the confidence interval, we use about two standard errors around the mean.

The next tutorial goes deeper into hypothesis testing. We can estimate confidence intervals around the mean from a sample using a one side \(t\)-test:

r3.(OneSampleTTest(wcgs.height) |> confint)
(177.011, 177.46)
cis(wcgs.height)
(outcome = 177.23546607482547,
 lower = 177.01130762199216,
 upper = 177.45962452765878,)

We can also estimate 95% CI by the method of bootstrap:

pubh.bst(wcgs.dbp) |> rcopy
1×5 DataFrame
Row stat estimate %CI lower upper
String Float64 Float64 Float64 Float64
1 mean 82.02 95.0 81.7 82.33

What about confidence intervals around the geometric mean?

pubh.bst(wcgs.dbp, stat="gmean") |> rcopy
1×5 DataFrame
Row stat estimate %CI lower upper
String Float64 Float64 Float64 Float64
1 gmean 81.47 95.0 81.14 81.8

And even around the median:

pubh.bst(wcgs.dbp, stat="median") |> rcopy
1×5 DataFrame
Row stat estimate %CI lower upper
String Float64 Float64 Float64 Float64
1 median 80.0 95.0 80.0 80.0

3.1 Stratified CIs

kfm = read_rds("data/kfm.rds") |> rcopy
kfm |> schema
┌────────────┬───────────────┬──────────────────────────────────┐
│ names      │ scitypes      │ types                            │
├────────────┼───────────────┼──────────────────────────────────┤
│ no         │ Count         │ Int64                            │
│ dl_milk    │ Continuous    │ Float64                          │
│ sex        │ Multiclass{2} │ CategoricalValue{String, UInt32} │
│ weight     │ Continuous    │ Float64                          │
│ ml_suppl   │ Count         │ Int64                            │
│ mat_weight │ Count         │ Int64                            │
│ mat_height │ Count         │ Int64                            │
└────────────┴───────────────┴──────────────────────────────────┘
combine(groupby(kfm, :sex), :dl_milk => cis => AsTable)
2×4 DataFrame
Row sex outcome lower upper
Cat… Float64 Float64 Float64
1 Boy 7.9524 7.36768 8.53712
2 Girl 7.0564 6.49906 7.61374
combine(groupby(wcgs |> dropmissing, [:chd, :dib_pat]), :chol => cis => AsTable)
4×5 DataFrame
Row chd dib_pat outcome lower upper
Cat… Cat… Float64 Float64 Float64
1 No CHD A 226.27 224.044 228.497
2 No CHD B 222.352 220.222 224.481
3 CHD A 248.887 241.275 256.499
4 CHD B 252.282 242.377 262.187

3.2 Confidence intervals for proportions

We can use the Binomial test for a proportion. We need the number of successes and the total number of observations.

tabulate(wcgs, :chd)
       | Freq. | Percent |  Cum.   
-----------------------------------
No CHD |  2897 |  91.852 |  91.852 
   CHD |   257 |   8.148 | 100.000 
-----------------------------------
 Total |  3154 | 100.000 |         
r3.(BinomialTest(257, nrow(wcgs)) |> confint)
(0.072, 0.092)